State and proof bernaulli's theorem

tatement: For the streamline flow of non-viscous and incompressible liquid, the sum of potential energy, kinetic energy and pressure energy is constant. 

Proof: 
 

Let us consider the ideal liquid of density ρ flowing through the pipe LM of varying cross-section.

Let P₁ and P₂ be the pressures at ends L and M and A₁ and A₂ be the areas of cross-sections at ends L and M respectively.

Let the liquid enter with velocity V₁ and leave with velocity v₂. 

Let A₁ > A₂. 

Now, by equation of continuity,

                      

Since, A₁ > A₂

Therefore, 

v₂ > v₁ and P₁ > P₂ 

Let, m be the mass of liquid enetring at end L in time t. 

The liquid will cover a distance = v₁t 

Therefore, the work done by pressure on the liquid at end L in time t is, 

W₁ = Force  

      = P₁A₁v₁t           ... (1) 

Since same mass m leaves the pipe at end M in same time t, in which liquid will cover the distance given by v₂t.

Therefore, work done by liquid against the force due to pressure P₁ is given by, 
 
        W₂ = P₂A₂v₂t        ... (2) 

Net ork done by pressure on the liquid in time t is, 

W = W₁ - W₂ 

    =P₁A₁v₁t  - P₂A₂v₂t        ... (3) 

This work done on liquid by pressure increases it's kinetic energy and potential energy. 

Increase in K.E of liquid is, 


That is,